cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A110665 Sequence is {a(0,n)}, where a(m,0)=0, a(m,n) = a(m-1,n)+a(m,n-1) and a(0,n) is such that a(n,n) = n for all n.

Original entry on oeis.org

0, 1, 0, -3, -4, 0, 6, 7, 0, -9, -10, 0, 12, 13, 0, -15, -16, 0, 18, 19, 0, -21, -22, 0, 24, 25, 0, -27, -28, 0, 30, 31, 0, -33, -34, 0, 36, 37, 0, -39, -40, 0, 42, 43, 0, -45, -46, 0, 48, 49, 0, -51, -52, 0, 54, 55, 0, -57, -58, 0, 60, 61, 0, -63, -64, 0, 66, 67, 0, -69, -70, 0, 72, 73, 0, -75, -76, 0, 78, 79
Offset: 0

Views

Author

Leroy Quet, Aug 02 2005

Keywords

Examples

			a(0,n): 0, 1, 0, -3, -4,...
a(1,n): 0, 1, 1, -2, -6,...
a(2,n): 0, 1, 2, 0, -6,...
a(3,n): 0, 1, 3, 3, -3,...
a(4,n): 0, 1, 4, 7, 4,...
Main diagonal of array is 0, 1, 2, 3, 4,...

Links

Crossrefs

Cf. A110666 - A110672.

Programs

  • Mathematica
    LinearRecurrence[{2,-3,2,-1},{0,1,0,-3},80] (* Harvey P. Dale, Dec 19 2015 *)
a[0, n_] := a[0, n] = If[n < 3, {0, 1, 0}[[n+1]], (n((n-2)a[0, n-1] - (n-1)a[0, n-2]))/((n-1)(n-2))];
Table[a[0, n], {n, 0, 79}] (* Jean-François Alcover, Mar 29 2020 *)
  • PARI
    my(x='x+O('x^50)); concat([0], Vec((x-2*x^2)/(1-x+x^2)^2)) \\ G. C. Greubel, Sep 03 2017

Formula

a(0, n) = n - Sum_{k=0..(n-1)} binomial(2*n-k-1, n-1)*a(0, k).
From Franklin T. Adams-Watters, May 12 2006: (Start)
a(n) = n * A010892(n), where A010892 is periodic sequence [1,1,0,-1,-1,0].
G.f.: (x-2*x^2)/(1-x+x^2)^2. (End)

Extensions

More terms from Franklin T. Adams-Watters, May 12 2006

A110666 Sequence is {a(1,n)}, where a(m,n) is defined at sequence A110665.

Original entry on oeis.org

0, 1, 1, -2, -6, -6, 0, 7, 7, -2, -12, -12, 0, 13, 13, -2, -18, -18, 0, 19, 19, -2, -24, -24, 0, 25, 25, -2, -30, -30, 0, 31, 31, -2, -36, -36, 0, 37, 37, -2, -42, -42, 0, 43, 43, -2, -48, -48, 0, 49, 49, -2, -54, -54, 0, 55, 55, -2, -60, -60, 0, 61, 61, -2, -66, -66, 0, 67, 67, -2, -72, -72, 0, 73, 73, -2, -78, -78, 0, 79, 79, -2, -84
Offset: 0

Views

Author

Leroy Quet, Aug 02 2005

Keywords

Examples

			a(0,n): 0,  1,  0, -3, -4, ...
a(1,n): 0,  1,  1, -2, -6, ...
a(2,n): 0,  1,  2,  0, -6, ...
a(3,n): 0,  1,  3,  3, -3, ...
a(4,n): 0,  1,  4,  7,  4, ...
Main diagonal of array is 0, 1, 2, 3, 4, ...

Links

Crossrefs

Cf. A110665, A110667, A110668, A110669, A110670, A110671, A110672.

Programs

  • Maple
    A11066x := proc(mmax,nmax) local a,i,j ; a := array(0..mmax,0..nmax) ; a[0,0] := 0 ; for i from 1 to nmax do a[0,i] := i-sum(binomial(2*i-k-1,i-1)*a[0,k],k=0..i-1) : od ; for j from 1 to mmax do a[j,0] := 0 ; for i from 1 to nmax do a[j,i] := a[j-1,i]+a[j,i-1] ; od ; od ; RETURN(a) ; end : nmax := 100 : m := 1: a := A11066x(m,nmax) : for n from 0 to nmax do printf("%d,",a[m,n]) ; od ; # R. J. Mathar, Sep 01 2006
  • Mathematica
    a[m_, 0] := 0; a[n_, n_] := n; a[0, n_] := n - Sum[Binomial[2*n - k - 1, n - 1]* a[0, k], {k, 0, n - 1}]; a[m_, n_] := a[m, n] = a[m - 1, n] + a[m, n - 1]; Table[a[1, n], {n, 0, 50}] (* G. C. Greubel, Sep 03 2017 *)

Formula

From R. J. Mathar, Oct 09 2013: (Start)
Conjecture: G.f. x*(-1+2*x) / ( (x-1)*(x^2-x+1)^2 ).
a(n) = -A010892(n-1) + A165202(n) -1. (End)

Extensions

More terms from R. J. Mathar, Sep 01 2006

A110667 Sequence is {a(2,n)}, where a(m,n) is defined at sequence A110665.

Original entry on oeis.org

0, 1, 2, 0, -6, -12, -12, -5, 2, 0, -12, -24, -24, -11, 2, 0, -18, -36, -36, -17, 2, 0, -24, -48, -48, -23, 2, 0, -30, -60, -60, -29, 2, 0, -36, -72, -72, -35, 2, 0, -42, -84, -84, -41, 2, 0, -48, -96, -96, -47, 2, 0, -54, -108, -108, -53, 2, 0, -60, -120, -120, -59, 2, 0, -66, -132, -132, -65, 2, 0, -72, -144, -144, -71, 2, 0
Offset: 0

Views

Author

Leroy Quet, Aug 02 2005

Keywords

Examples

			a(0,n): 0,  1,  0, -3, -4, ...
a(1,n): 0,  1,  1, -2, -6, ...
a(2,n): 0,  1,  2,  0, -6, ...
a(3,n): 0,  1,  3,  3, -3, ...
a(4,n): 0,  1,  4,  7,  4, ...
Main diagonal of array is 0, 1, 2, 3, 4, ...

Links

Crossrefs

Cf. A110665 - A110672.

Programs

  • Maple
    A11066x := proc(mmax,nmax) local a,i,j ; a := array(0..mmax,0..nmax) ; a[0,0] := 0 ; for i from 1 to nmax do a[0,i] := i-sum(binomial(2*i-k-1,i-1)*a[0,k],k=0..i-1) : od ; for j from 1 to mmax do a[j,0] := 0 ; for i from 1 to nmax do a[j,i] := a[j-1,i]+a[j,i-1] ; od ; od ; RETURN(a) ; end :
nmax := 100 : m := 2: a := A11066x(m,nmax) :
for n from 0 to nmax do printf("%d,",a[m,n]) ; od ; # R. J. Mathar, Sep 01 2006
  • Mathematica
    a[m_, n_] := a[m, n] = Which[n == 0, 0, m == 0, n - Sum[ Binomial[2 n - k - 1, n - 1]*a[0, k], {k, 0, (n - 1)}], True, a[m - 1, n] + a[m, n - 1]]; Array[a[2, #] &, 76, 0] (* Michael De Vlieger, Sep 04 2017 *)

Formula

Conjecture: g.f.: -x*(-1+2*x) / ( (x-1)^2*(x^2-x+1)^2 ). - R. J. Mathar, Oct 09 2013

Extensions

More terms from R. J. Mathar, Sep 01 2006

A110668 Sequence is {a(3,n)}, where a(m,n) is defined at sequence A110665.

Original entry on oeis.org

0, 1, 3, 3, -3, -15, -27, -32, -30, -30, -42, -66, -90, -101, -99, -99, -117, -153, -189, -206, -204, -204, -228, -276, -324, -347, -345, -345, -375, -435, -495, -524, -522, -522, -558, -630, -702, -737, -735, -735, -777, -861, -945, -986, -984, -984, -1032, -1128, -1224, -1271, -1269, -1269, -1323, -1431
Offset: 0

Views

Author

Leroy Quet, Aug 02 2005

Keywords

Examples

			a(0,n): 0,  1,  0, -3, -4, ...
a(1,n): 0,  1,  1, -2, -6, ...
a(2,n): 0,  1,  2,  0, -6, ...
a(3,n): 0,  1,  3,  3, -3, ...
a(4,n): 0,  1,  4,  7,  4, ...
Main diagonal of array is 0, 1, 2, 3, 4, ...

Links

Crossrefs

Cf. A110665 - A110672.

Programs

  • Maple
    A11066x := proc(mmax,nmax) local a,i,j ; a := array(0..mmax,0..nmax) ; a[0,0] := 0 ; for i from 1 to nmax do a[0,i] := i-sum(binomial(2*i-k-1,i-1)*a[0,k],k=0..i-1) : od ; for j from 1 to mmax do a[j,0] := 0 ; for i from 1 to nmax do a[j,i] := a[j-1,i]+a[j,i-1] ; od ; od ; RETURN(a) ; end :
nmax := 100 : m := 3: a := A11066x(m,nmax) :
for n from 0 to nmax do printf("%d,",a[m,n]) ; od ; # R. J. Mathar, Sep 01 2006
  • Mathematica
    a[m_, n_] := a[m, n] = Which[n == 0, 0, m == 0, n - Sum[ Binomial[2 n - k - 1, n - 1]*a[0, k], {k, 0, (n - 1)}], True, a[m - 1, n] + a[m, n - 1]]; Array[a[3, #] &, 54, 0] (* Michael De Vlieger, Sep 04 2017 *)

Formula

Conjecture: g.f.: x*(-1+2*x) / ( (x^2-x+1)^2*(x-1)^3 ). - R. J. Mathar, Oct 09 2013

Extensions

More terms from R. J. Mathar, Sep 01 2006

A110669 Sequence is {a(4,n)}, where a(m,n) is defined at sequence A110665.

Original entry on oeis.org

0, 1, 4, 7, 4, -11, -38, -70, -100, -130, -172, -238, -328, -429, -528, -627, -744, -897, -1086, -1292, -1496, -1700, -1928, -2204, -2528, -2875, -3220, -3565, -3940, -4375, -4870, -5394, -5916, -6438, -6996, -7626, -8328, -9065, -9800, -10535, -11312, -12173, -13118, -14104, -15088, -16072, -17104
Offset: 0

Views

Author

Leroy Quet, Aug 02 2005

Keywords

Examples

			a(0,n): 0, 1, 0, -3, -4,...
a(1,n): 0, 1, 1, -2, -6,...
a(2,n): 0, 1, 2, 0, -6,...
a(3,n): 0, 1, 3, 3, -3,...
a(4,n): 0, 1, 4, 7, 4,...
Main diagonal of array is 0, 1, 2, 3, 4,...

Crossrefs

Cf. A110665, A110666, A110667, A110668, A110670, A110671, A110672.

Programs

  • Maple
    A11066x := proc(mmax,nmax) local a,i,j ; a := array(0..mmax,0..nmax) ; a[0,0] := 0 ; for i from 1 to nmax do a[0,i] := i-sum(binomial(2*i-k-1,i-1)*a[0,k],k=0..i-1) : od ; for j from 1 to mmax do a[j,0] := 0 ; for i from 1 to nmax do a[j,i] := a[j-1,i]+a[j,i-1] ; od ; od ; RETURN(a) ; end : nmax := 100 : m := 4: a := A11066x(m,nmax) : for n from 0 to nmax do printf("%d,",a[m,n]) ; od ; # R. J. Mathar, Sep 01 2006
  • Mathematica
    a[_, 0] = 0;
a[0, n_] := a[0, n] = If[n < 3, {0, 1, 0}[[n+1]], (n((n-2) a[0, n-1] - (n-1)a[0, n-2]))/((n-1)(n-2))];
a[m_, n_] := a[m, n] = a[m-1, n] + a[m, n-1];
Table[a[4, n], {n, 0, 46}] (* Jean-François Alcover, Mar 29 2020 *)

Formula

Empirical g.f.: -x*(2*x-1) / ((x-1)^4*(x^2-x+1)^2). - Colin Barker, Jul 02 2014

Extensions

More terms from R. J. Mathar, Sep 01 2006

A110670 Sequence is {a(5,n)}, where a(m,n) is defined at sequence A110665.

Original entry on oeis.org

0, 1, 5, 12, 16, 5, -33, -103, -203, -333, -505, -743, -1071, -1500, -2028, -2655, -3399, -4296, -5382, -6674, -8170, -9870, -11798, -14002, -16530, -19405, -22625, -26190, -30130, -34505, -39375, -44769, -50685, -57123, -64119, -71745, -80073, -89138, -98938, -109473, -120785, -132958
Offset: 0

Views

Author

Leroy Quet, Aug 02 2005

Keywords

Examples

			a(0,n): 0, 1, 0, -3, -4,...
a(1,n): 0, 1, 1, -2, -6,...
a(2,n): 0, 1, 2, 0, -6,...
a(3,n): 0, 1, 3, 3, -3,...
a(4,n): 0, 1, 4, 7, 4,...
Main diagonal of array is 0, 1, 2, 3, 4,...

Crossrefs

Cf. A110665, A110666, A110667, A110668, A110669, A110671, A110672.

Programs

  • Maple
    A11066x := proc(mmax,nmax) local a,i,j ; a := array(0..mmax,0..nmax) ; a[0,0] := 0 ; for i from 1 to nmax do a[0,i] := i-sum(binomial(2*i-k-1,i-1)*a[0,k],k=0..i-1) : od ; for j from 1 to mmax do a[j,0] := 0 ; for i from 1 to nmax do a[j,i] := a[j-1,i]+a[j,i-1] ; od ; od ; RETURN(a) ; end : nmax := 100 : m := 5: a := A11066x(m,nmax) : for n from 0 to nmax do printf("%d,",a[m,n]) ; od ; # R. J. Mathar, Sep 01 2006
  • Mathematica
    a[_, 0] = 0;
a[0, n_] := a[0, n] = If[n < 3, {0, 1, 0}[[n+1]], (n((n-2)a[0, n-1] - (n-1)a[0, n-2]))/((n-1)(n-2))];
a[m_, n_] := a[m, n] = a[m-1, n] + a[m, n-1];
Table[a[5, n], {n, 0, 41}] (* Jean-François Alcover, Mar 29 2020 *)

Formula

Empirical g.f.: x*(2*x-1) / ((x-1)^5*(x^2-x+1)^2). - Colin Barker, Jul 02 2014

Extensions

More terms from R. J. Mathar, Sep 01 2006

A110671 Sequence is {a(6,n)}, where a(m,n) is defined at sequence A110665.

Original entry on oeis.org

0, 1, 6, 18, 34, 39, 6, -97, -300, -633, -1138, -1881, -2952, -4452, -6480, -9135, -12534, -16830, -22212, -28886, -37056, -46926, -58724, -72726, -89256, -108661, -131286, -157476, -187606, -222111, -261486, -306255, -356940, -414063, -478182, -549927, -630000, -719138, -818076
Offset: 0

Views

Author

Leroy Quet, Aug 02 2005

Keywords

Examples

			a(0,n): 0, 1, 0, -3, -4,...
a(1,n): 0, 1, 1, -2, -6,...
a(2,n): 0, 1, 2, 0, -6,...
a(3,n): 0, 1, 3, 3, -3,...
a(4,n): 0, 1, 4, 7, 4,...
Main diagonal of array is 0, 1, 2, 3, 4,...

Crossrefs

Cf. A110665, A110666, A110667, A110668, A110669, A110670, A110672.

Programs

  • Maple
    A11066x := proc(mmax,nmax) local a,i,j ; a := array(0..mmax,0..nmax) ; a[0,0] := 0 ; for i from 1 to nmax do a[0,i] := i-sum(binomial(2*i-k-1,i-1)*a[0,k],k=0..i-1) : od ; for j from 1 to mmax do a[j,0] := 0 ; for i from 1 to nmax do a[j,i] := a[j-1,i]+a[j,i-1] ; od ; od ; RETURN(a) ; end : nmax := 100 : m := 6: a := A11066x(m,nmax) : for n from 0 to nmax do printf("%d,",a[m,n]) ; od ; # R. J. Mathar, Sep 01 2006
  • Mathematica
    a[_, 0] = 0;
a[0, n_] := a[0, n] = If[n < 3, {0, 1, 0}[[n+1]], (n((n-2)a[0, n-1] - (n-1)a[0, n-2]))/((n-1)(n-2))];
a[m_, n_] := a[m, n] = a[m-1, n] + a[m, n-1];
Table[a[6, n], {n, 0, 38}] (* Jean-François Alcover, Mar 29 2020 *)

Formula

Empirical g.f.: -x*(2*x-1) / ((x-1)^6*(x^2-x+1)^2). - Colin Barker, Jul 02 2014

Extensions

More terms from R. J. Mathar, Sep 01 2006
Showing 1-7 of 7 results.

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